There was an outcry in New York theatre recently over the casting of a hearing actor in the role of John Singer, the deaf-mute character in a stage adaptation of Carson McCullers's The Heart Is a Lonely Hunter. Many deaf people believed that the role should have gone to a deaf actor, with Linda Bove, a deaf actor and board member of the Alliance for Inclusion in the Arts, raising the temperature of the debate by saying: "A hearing actor playing a deaf character is tantamount to putting a white actor in blackface". The production proved to be a reminder that the deaf community is still marginalised and finding its voice in a largely hearing theatre world.

Here in the UK, we have some extraordinary deaf theatremakers and performers, including Sophie Woolley (creator of the cult one-woman show, Born to Run, and currently appearing as Gabriella in C4's Cast Offs, a show about a group of disabled people marooned on an island), and the actor Caroline Parker, who will be in Signs of a Diva at the Theatre Royal Stratford East early next year (and who, over a long career, has tackled everything from comedy to the lead role in the 17th-century Jacobean revenge drama The Changeling). Even so, the stories of the deaf world are only just beginning to be heard.

One deaf-led company, Deafinitely Theatre, is doing its utmost to bring these stories centre-stage. "There are so many misconceptions about being deaf," says Deafinitely's artistic director, Paula Garfield. "We want people to understand what it means to be deaf, but also to see the potential of deaf artists." That second point may be the crucial one. Later this week, at London's Drill Hall, the company will be showing some of the work that has come out of its Deafinitely Creative initiative, a three-year programme that began in 2008, which is designed to train deaf playwrights to a professional level. The scheme is the first and only one of its kind in this country, and its success will be judged not just on its ability to play midwife to a generation of deaf playwrights, but on whether those writers can eventually find work in a hearing theatre world.

The plays will be written in English, then translated into British Sign Language – something that, for many deaf people, remains a problematic idea. "English is the language of oppression, of being forbidden to use sign language," argues Deafinitely's Tomato Lichy. For many years, deaf people were discouraged from learning to sign and instead told to concentrate their energies on learning to speak. As a result, many grew up isolated from both the hearing and the deaf, unable to communicate effectively with either.

Born to hearing parents who were advised by the medical profession that it was imperative that she not be taught to sign, Garfield felt that she was a failure by the time she was a teenager – as a profoundly deaf person, she failed to learn to speak in a way that could be understood in the hearing world. Lichy's story is similar: after dropping out of university in his early twenties because of difficulties doing a course designed for hearing students, he became depressed, then homeless. He says he only emerged from that dark period when he threw away his hearing aids and learned to sign. Several years on, the couple couldn't be happier, both as artists and as parents – they have a daughter, Molly, and a second baby on the way.

Not that this has made their work any less politically urgent. Nothing quite prepared me for Playing God, which opened at London's Soho theatre in 2007. Inspired by their own experiences dealing with doctors who prized being able to hear over being able to communicate, it concerned a fictional couple at loggerheads over whether their deaf child should be fitted with cochlear implants. The story was very close to home; Molly is also deaf.

The use of implants is controversial in the deaf community because they don't restore hearing, and many believe that in any case deafness should not be "fixed". The play's argument was most powerfully voiced by the child's deaf father, who, when his daughter is fitted with implants, talks of watching "as my daughter's deafness died". But while Playing God set out to provoke discussion over the use of implants – and did so effectively – there was no doubt that the emotional argument to not fit the implants was biased in favour of the father's viewpoint. It was a sad, angry play , and one that seemed to me extraordinarily one-sided. .

The couple had plenty of reason to feel angry, says Lichy: "When our daughter was labelled a problem because she couldn't hear, we were shocked. The doctors were only interested in how she could be repaired. Repaired! I told them, 'My daughter doesn't need repairing.'" "We've moved a long way since Playing God," says Garfield. "I was a very angry woman around that time. I'm not any more."

These stories are important, but over time, different narratives will appear too. Deafinitely is starting to develop a distinctive approach, combining the visual and physical, and not just producing issue-based theatre. The director of Soho's writers' centre, Nina Steiger, agrees. "The company is starting to think in theatre, and that's a good sign." She suggests that deaf theatre has sometimes been too focused on narrow political concerns. "There is a lot of dogma and politics in the deaf theatre world," she says. "It can veer towards being isolationist, and I think that Playing God did have a tendency towards that." But there's nothing new in that; many of the most exciting theatre companies of the last 30 years have built on personal experience. "Back in the 1970s, in gay theatre, it was the coming-out narrative that dominated," Steiger argues. "Today the deaf equivalent is throwing off your hearing aids and the articulation of a deaf world that exists independently of a hearing world."

If society in general is biased towards the spoken word, then British theatre tradition has been even more so. Arguably, it's only in the last few years that theatre in this country has woken up to the potential that visual and physical techniques can bring to productions of all kinds. So what's next for deaf theatre? Jenny Sealey, herself a deaf artist, and the artistic director of the pioneering disabled-led theatre company Graeae, founded in 1980 by Nabil Shaban and Richard Tomlinson and which makes work that ranges from classical revivals to street arts, would like to see companies doing shows entirely in British Sign Language. Perhaps some Shakespeare, Beckett or even a big musical like La Cage aux Folles.

Certainly, British Sign Language is wonderfully visual, and has layers of metaphor that make English seem unexpressive and unwieldy. It is also economical in terms of its visual language, something that Steiger believes playwrights writing in English could learn from. Playwright Andrew Muir, who works as a dramaturg with Deafinitely and wrote Double Sentence, a play about a young deaf man sent to prison for 12 years who faces problems because of a lack of deaf access in the penal system, for Deafinitely concurs: "It took a week [for my play] to be translated from English to British Sign Language, and it made me realise just how much bullshit I write, and how much could be cut."

Garfield is in no doubt that Deafinitely still has a long way to go, though, and that this week's showcase of new plays by deaf writers is only a staging post along the way. "The rest of the theatre world has had hundreds of years to learn," she says. "We are beginners – but we're catching up fast."